Is ${548105}$ divisible by $3$ ?
Explanation: A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {548105}= &&{5}\cdot100000+ \\&&{4}\cdot10000+ \\&&{8}\cdot1000+ \\&&{1}\cdot100+ \\&&{0}\cdot10+ \\&&{5}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {548105}= &&{5}(99999+1)+ \\&&{4}(9999+1)+ \\&&{8}(999+1)+ \\&&{1}(99+1)+ \\&&{0}(9+1)+ \\&&{5} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {548105}= &&\gray{5\cdot99999}+ \\&&\gray{4\cdot9999}+ \\&&\gray{8\cdot999}+ \\&&\gray{1\cdot99}+ \\&&\gray{0\cdot9}+ \\&& {5}+{4}+{8}+{1}+{0}+{5} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${548105}$ is divisible by $3$ if ${ 5}+{4}+{8}+{1}+{0}+{5}$ is divisible by $3$ Add the digits of ${548105}$ $ {5}+{4}+{8}+{1}+{0}+{5} = {23} $ If ${23}$ is divisible by $3$ , then ${548105}$ must also be divisible by $3$ ${23}$ is not divisible by $3$, therefore ${548105}$ must not be divisible by $3$.